The kernels of radical homomorphisms and intersections of prime ideals

Abstract

We establish a necessary condition for a commutative Banach algebra A A so that there exists a homomorphism θ \theta from A A into another Banach algebra such that the prime radical of the continuity ideal of θ \theta is not a finite intersection of prime ideals in A A . We prove that the prime radical of the continuity ideal of an epimorphism from A A onto another Banach algebra (or of a derivation from A A into a Banach A A -bimodule) is always a finite intersection of prime ideals. Under an additional cardinality condition (and assuming the Continuum Hypothesis), this necessary condition is proved to be sufficient. En route, we give a general result on norming commutative semiprime algebras; extending the class of algebras known to be normable. We characterize those locally compact metrizable spaces Ω \Omega for which there exists a homomorphism from C 0 ( Ω ) \mathcal C_0(\Omega ) into a radical Banach algebra whose kernel is not a finite intersection of prime ideals.